Problem 2
Question
Fill in the blanks. We read \((x+7)(2 x-3)\) as "the _____ of \(x+7\) _____ the quantity of \(2 x-3\) "
Step-by-Step Solution
Verified Answer
Product; and
1Step 1: Understand Expressions
The expression \( (x+7)(2x-3) \) represents two binomials being multiplied together. The expression inside each parenthesis is called a 'quantity.' \( x+7 \) is the first quantity, and \( 2x-3 \) is the second.
2Step 2: Identify Mathematical Operation
The operation occurring between the two binomials is multiplication. In mathematics, when two expressions are next to each other (such as \( (x+7) \) and \( (2x-3) \)), it's implied that these expressions should be multiplied.
3Step 3: Fill in the Blanks
We read the expression \( (x+7)(2x-3) \) as "the product of \( x+7 \) and the quantity of \( 2x-3 \)." 'Product' denotes multiplication, and 'and' signals the combination of the two quantities in multiplication.
Key Concepts
Understanding BinomialsExploring Multiplication of BinomialsPerforming Mathematical Operations with Binomials
Understanding Binomials
In algebra, a binomial is an expression that consists of two terms connected by a plus or minus sign. Binomials are a crucial component in both basic and advanced algebra. They often form the foundation for more complex mathematical operations and are widely used in equations and polynomial expressions. For instance, the expression \(x+7\) is a binomial because it combines two terms, \(x\) and 7, using a plus sign. Likewise, \(2x-3\) is a binomial with the terms \(2x\) and -3, combined with a minus sign.
When working with binomials, it's important to recognize each part. Doing so allows you to apply the correct mathematical operations, like addition, subtraction, or multiplication, to solve problems. Understanding binomials is essential because they frequently appear in a wide range of algebraic expressions. Binomials form a significant part of learning how to manipulate expressions and solve equations across various mathematical fields.
When working with binomials, it's important to recognize each part. Doing so allows you to apply the correct mathematical operations, like addition, subtraction, or multiplication, to solve problems. Understanding binomials is essential because they frequently appear in a wide range of algebraic expressions. Binomials form a significant part of learning how to manipulate expressions and solve equations across various mathematical fields.
Exploring Multiplication of Binomials
Multiplication is a basic but powerful operation in mathematics, especially when applied to binomials. Multiplying binomials involves expanding the expressions to form a quadratic expression. This process is often carried out using methods such as the distributive property or the FOIL method. FOIL, which stands for First, Outer, Inner, Last refers to the procedure for multiplying each term in the first binomial with each term in the second.
To illustrate, consider the binomials \((x+7)\) and \((2x-3)\). When multiplied, you first take the 'First' terms, \(x\) and \(2x\), multiplying them to get \(2x^2\). Next, tackle the 'Outer' pair, \(x\) and -3, yielding \(-3x\). Then, focus on the 'Inner' terms, 7 and \(2x\), leading to \(14x\). Lastly, multiply the 'Last' terms, 7 and -3, resulting in -21.
Combining these results, you obtain the expression \(2x^2 + 11x - 21\). Multiplying binomials not only helps to simplify expressions but also paves the way for further mathematical applications, such as factoring or solving equations.
To illustrate, consider the binomials \((x+7)\) and \((2x-3)\). When multiplied, you first take the 'First' terms, \(x\) and \(2x\), multiplying them to get \(2x^2\). Next, tackle the 'Outer' pair, \(x\) and -3, yielding \(-3x\). Then, focus on the 'Inner' terms, 7 and \(2x\), leading to \(14x\). Lastly, multiply the 'Last' terms, 7 and -3, resulting in -21.
Combining these results, you obtain the expression \(2x^2 + 11x - 21\). Multiplying binomials not only helps to simplify expressions but also paves the way for further mathematical applications, such as factoring or solving equations.
Performing Mathematical Operations with Binomials
Mathematical operations with binomials include addition, subtraction, multiplication, and sometimes division. The focus is often on how these operations change the structure of expressions. Multiplying binomials is a foundational concept in algebra, and mastering this skill is essential for success in more advanced mathematics.
When performing these operations, it's important to remember the order of operations (PEMDAS/BODMAS). This rule ensures the correct sequence is followed, maintaining the integrity of the mathematical expression. For binomials, however, the focus is primarily on multiplication, where each term of one binomial is multiplied with each term of another, producing a new polynomial.
Operations with binomials enhance problem-solving skills by encouraging logical thinking and an understanding of algebraic structures. They also provide the foundational skills necessary for exploring more complex concepts like polynomial division and algebraic fractions. Exploring these operations doesn't just hone algebraic skills but also supports broader mathematical comprehension, helping students handle diverse problems with confidence.
When performing these operations, it's important to remember the order of operations (PEMDAS/BODMAS). This rule ensures the correct sequence is followed, maintaining the integrity of the mathematical expression. For binomials, however, the focus is primarily on multiplication, where each term of one binomial is multiplied with each term of another, producing a new polynomial.
Operations with binomials enhance problem-solving skills by encouraging logical thinking and an understanding of algebraic structures. They also provide the foundational skills necessary for exploring more complex concepts like polynomial division and algebraic fractions. Exploring these operations doesn't just hone algebraic skills but also supports broader mathematical comprehension, helping students handle diverse problems with confidence.
Other exercises in this chapter
Problem 1
Fill in the blank. Expressions such as \(x^{4}, 10^{3},\) and \((5 t)^{2}\) are called _____ expressions.
View solution Problem 2
Fill in the blanks. \((2 x+3)^{2}\) is the _____ of a binomial and \((a+6)(a-6)\) is the product of the sum and difference of the _____ two terms.
View solution Problem 2
\(\left(b^{2}-9 b+11\right)-\left(4 b^{2}-14 b\right)\) is the _______ of a trinomial and a binomial.
View solution Problem 2
Fill in the blanks. The ____ of a polynomial are separated by \(+\) symbols.
View solution